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Theorem ltexprlemloc 7429
Description: Our constructed difference is located. Lemma for ltexpri 7435. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemloc  |-  ( A 
<P  B  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemloc
Dummy variables  z  w  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 7231 . . . . . 6  |-  ( q 
<Q  r  ->  E. w  e.  Q.  ( q  +Q  w )  =  r )
21adantl 275 . . . . 5  |-  ( ( A  <P  B  /\  q  <Q  r )  ->  E. w  e.  Q.  ( q  +Q  w
)  =  r )
3 ltrelpr 7327 . . . . . . . . . 10  |-  <P  C_  ( P.  X.  P. )
43brel 4591 . . . . . . . . 9  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
54simpld 111 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
6 prop 7297 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
7 prarloc 7325 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
86, 7sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
95, 8sylan 281 . . . . . . 7  |-  ( ( A  <P  B  /\  w  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) y  <Q  ( z  +Q  w ) )
109ad2ant2r 500 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w ) )
114simprd 113 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
1211ad2antrr 479 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  B  e.  P. )
1312ad2antrr 479 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  B  e.  P. )
14 ltanqg 7222 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1514adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  -> 
( f  <Q  g  <->  ( h  +Q  f ) 
<Q  ( h  +Q  g
) ) )
16 elprnqu 7304 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
176, 16sylan 281 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
185, 17sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
1918adantlr 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2019ad2ant2rl 502 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  y  e.  Q. )
21 elprnql 7303 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
226, 21sylan 281 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
235, 22sylan 281 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2423adantlr 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2524ad2ant2r 500 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  z  e.  Q. )
26 simplrl 524 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  w  e.  Q. )
27 addclnq 7197 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  +Q  w
)  e.  Q. )
2825, 26, 27syl2anc 408 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  w )  e.  Q. )
29 ltrelnq 7187 . . . . . . . . . . . . . . . . . . 19  |-  <Q  C_  ( Q.  X.  Q. )
3029brel 4591 . . . . . . . . . . . . . . . . . 18  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
3130simpld 111 . . . . . . . . . . . . . . . . 17  |-  ( q 
<Q  r  ->  q  e. 
Q. )
3231adantl 275 . . . . . . . . . . . . . . . 16  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
q  e.  Q. )
3332ad2antrr 479 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  q  e.  Q. )
34 addcomnqg 7203 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3534adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3615, 20, 28, 33, 35caovord2d 5940 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  <->  ( y  +Q  q )  <Q  (
( z  +Q  w
)  +Q  q ) ) )
37 addassnqg 7204 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  q  e.  Q. )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  ( w  +Q  q
) ) )
3825, 26, 33, 37syl3anc 1216 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  ( w  +Q  q
) ) )
39 addcomnqg 7203 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e.  Q.  /\  q  e.  Q. )  ->  ( w  +Q  q
)  =  ( q  +Q  w ) )
4026, 33, 39syl2anc 408 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
w  +Q  q )  =  ( q  +Q  w ) )
4140oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  ( w  +Q  q ) )  =  ( z  +Q  ( q  +Q  w
) ) )
42 simplrr 525 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
q  +Q  w )  =  r )
4342oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
z  +Q  ( q  +Q  w ) )  =  ( z  +Q  r ) )
4438, 41, 433eqtrd 2176 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( z  +Q  w
)  +Q  q )  =  ( z  +Q  r ) )
4544breq2d 3941 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
( y  +Q  q
)  <Q  ( ( z  +Q  w )  +Q  q )  <->  ( y  +Q  q )  <Q  (
z  +Q  r ) ) )
4636, 45bitrd 187 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  <->  ( y  +Q  q )  <Q  (
z  +Q  r ) ) )
4746biimpa 294 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  r
) )
48 prop 7297 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
49 prloc 7313 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  <Q  ( z  +Q  r ) )  -> 
( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
5048, 49sylan 281 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  <Q  ( z  +Q  r ) )  -> 
( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
5113, 47, 50syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  ( z  e.  ( 1st `  A
)  /\  y  e.  ( 2nd `  A ) ) )  /\  y  <Q  ( z  +Q  w
) )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) )
5251ex 114 . . . . . . . . . 10  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  ( z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A ) ) )  ->  (
y  <Q  ( z  +Q  w )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) ) )
5352anassrs 397 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  q  <Q  r )  /\  (
w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  /\  z  e.  ( 1st `  A
) )  /\  y  e.  ( 2nd `  A
) )  ->  (
y  <Q  ( z  +Q  w )  ->  (
( y  +Q  q
)  e.  ( 1st `  B )  \/  (
z  +Q  r )  e.  ( 2nd `  B
) ) ) )
5453reximdva 2534 . . . . . . . 8  |-  ( ( ( ( A  <P  B  /\  q  <Q  r
)  /\  ( w  e.  Q.  /\  ( q  +Q  w )  =  r ) )  /\  z  e.  ( 1st `  A ) )  -> 
( E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
5554reximdva 2534 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
56 prml 7299 . . . . . . . . . . . 12  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. z  e.  Q.  z  e.  ( 1st `  A ) )
57 rexex 2479 . . . . . . . . . . . 12  |-  ( E. z  e.  Q.  z  e.  ( 1st `  A
)  ->  E. z 
z  e.  ( 1st `  A ) )
586, 56, 573syl 17 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  E. z 
z  e.  ( 1st `  A ) )
59 r19.45mv 3456 . . . . . . . . . . 11  |-  ( E. z  z  e.  ( 1st `  A )  ->  ( E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
605, 58, 593syl 17 . . . . . . . . . 10  |-  ( A 
<P  B  ->  ( E. z  e.  ( 1st `  A ) ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
6160adantr 274 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
62 prmu 7300 . . . . . . . . . . . . 13  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  A ) )
63 rexex 2479 . . . . . . . . . . . . 13  |-  ( E. x  e.  Q.  x  e.  ( 2nd `  A
)  ->  E. x  x  e.  ( 2nd `  A ) )
646, 62, 633syl 17 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  E. x  x  e.  ( 2nd `  A ) )
65 r19.9rmv 3454 . . . . . . . . . . . . . 14  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( ( z  +Q  r )  e.  ( 2nd `  B
)  <->  E. y  e.  ( 2nd `  A ) ( z  +Q  r
)  e.  ( 2nd `  B ) ) )
6665orbi2d 779 . . . . . . . . . . . . 13  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. y  e.  ( 2nd `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
67 r19.43 2589 . . . . . . . . . . . . 13  |-  ( E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. y  e.  ( 2nd `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) )
6866, 67syl6rbbr 198 . . . . . . . . . . . 12  |-  ( E. x  x  e.  ( 2nd `  A )  ->  ( E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
695, 64, 683syl 17 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( E. y  e.  ( 2nd `  A ) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
7069rexbidv 2438 . . . . . . . . . 10  |-  ( A 
<P  B  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
7170adantr 274 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  E. z  e.  ( 1st `  A
) ( E. y  e.  ( 2nd `  A
) ( y  +Q  q )  e.  ( 1st `  B )  \/  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
72 ibar 299 . . . . . . . . . . . . . . 15  |-  ( q  e.  Q.  ->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
7372adantr 274 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( E. y ( y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
74 ibar 299 . . . . . . . . . . . . . . 15  |-  ( r  e.  Q.  ->  ( E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) )  <->  ( r  e. 
Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
7574adantl 275 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( E. z ( z  e.  ( 1st `  A )  /\  (
z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( r  e.  Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
7673, 75orbi12d 782 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) ) )
7730, 76syl 14 . . . . . . . . . . . 12  |-  ( q 
<Q  r  ->  ( ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) ) )
78 ltexprlem.1 . . . . . . . . . . . . . 14  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
7978ltexprlemell 7420 . . . . . . . . . . . . 13  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
8078ltexprlemelu 7421 . . . . . . . . . . . . . 14  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
81 eleq1 2202 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
82 oveq1 5781 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
y  +Q  r )  =  ( z  +Q  r ) )
8382eleq1d 2208 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
( y  +Q  r
)  e.  ( 2nd `  B )  <->  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
8481, 83anbi12d 464 . . . . . . . . . . . . . . . 16  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
8584cbvexv 1890 . . . . . . . . . . . . . . 15  |-  ( E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) )  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
8685anbi2i 452 . . . . . . . . . . . . . 14  |-  ( ( r  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( r  e.  Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
8780, 86bitri 183 . . . . . . . . . . . . 13  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
8879, 87orbi12i 753 . . . . . . . . . . . 12  |-  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  \/  (
r  e.  Q.  /\  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
8977, 88syl6rbbr 198 . . . . . . . . . . 11  |-  ( q 
<Q  r  ->  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
90 df-rex 2422 . . . . . . . . . . . 12  |-  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  <->  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )
91 df-rex 2422 . . . . . . . . . . . 12  |-  ( E. z  e.  ( 1st `  A ) ( z  +Q  r )  e.  ( 2nd `  B
)  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) )
9290, 91orbi12i 753 . . . . . . . . . . 11  |-  ( ( E. y  e.  ( 2nd `  A ) ( y  +Q  q
)  e.  ( 1st `  B )  \/  E. z  e.  ( 1st `  A ) ( z  +Q  r )  e.  ( 2nd `  B
) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  \/  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9389, 92syl6bbr 197 . . . . . . . . . 10  |-  ( q 
<Q  r  ->  ( ( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9493adantl 275 . . . . . . . . 9  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) )  <->  ( E. y  e.  ( 2nd `  A ) ( y  +Q  q )  e.  ( 1st `  B
)  \/  E. z  e.  ( 1st `  A
) ( z  +Q  r )  e.  ( 2nd `  B ) ) ) )
9561, 71, 943bitr4rd 220 . . . . . . . 8  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) )  <->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
9695adantr 274 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( (
q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C
) )  <->  E. z  e.  ( 1st `  A
) E. y  e.  ( 2nd `  A
) ( ( y  +Q  q )  e.  ( 1st `  B
)  \/  ( z  +Q  r )  e.  ( 2nd `  B
) ) ) )
9755, 96sylibrd 168 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( E. z  e.  ( 1st `  A ) E. y  e.  ( 2nd `  A
) y  <Q  (
z  +Q  w )  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
9810, 97mpd 13 . . . . 5  |-  ( ( ( A  <P  B  /\  q  <Q  r )  /\  ( w  e.  Q.  /\  ( q  +Q  w
)  =  r ) )  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) )
992, 98rexlimddv 2554 . . . 4  |-  ( ( A  <P  B  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C ) ) )
10099ex 114 . . 3  |-  ( A 
<P  B  ->  ( q 
<Q  r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
101100ralrimivw 2506 . 2  |-  ( A 
<P  B  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
102101ralrimivw 2506 1  |-  ( A 
<P  B  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7102    +Q cplq 7104    <Q cltq 7107   P.cnp 7113    <P cltp 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iltp 7292
This theorem is referenced by:  ltexprlempr  7430
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